Spring 2008 #5

Problem Statement: Let $\{f_n\}$ be a sequence of continuous functions defined on $[a,b]$ and $f$ a real valued function defined on $[a,b]$. (a) Prove if $\{f_n\}\rightarrow f$ uniformly then $\lim\limits_{n\rightarrow \infty}\int\limits_{a}^{b}f_n(x)dx=\int\limits_{a}^{b}f(x)dx$ and (b) Does this hold is $\{f_n\}\rightarrow f$ pointwise? Prove or give a counter example.

(a) Proof: We are given that $f_n\rightarrow f$ uniformly and so since each $f_n$ is continuous it follows that $f$ is continuous since our convergence is uniform. Thus, $f$ is Riemann Integrable on $[a,b]$.

To prove the given statement we will prove an equivalent statement, that $\lim\limits_{n\rightarrow\infty}\int\limits_{a}^{b}(f_n(x)-f(x))dx=0$.

Let $\varepsilon>0$. Since $f_n\rightarrow f$ uniformly there exists some $N\in\mathbb{N}$ such that $|f_n(x)-f(x)|<\dfrac{\varepsilon}{b-a}$ for every $n\geq N$ and $x\in [a,b]$. Consider the following for $n\geq N$:

$|\int\limits_{a}^{b}(f_n(x)-f(x))dx|\leq \int_{a}^{b}|f_n(x)-f(x)|dx$

$<\int\limits_{a}^{b}\dfrac{\varepsilon}{b-a}dx$

$=\dfrac{\varepsilon}{b-a}(b-a)=\varepsilon$

Thus we have shown that $\lim\limits_{n\rightarrow\infty}\int\limits_{a}^{b}(f_n(x)-f(x))dx=0$ and so we have proven the original statement.

$\Box$

(b) Solution: Claim, this does not hold if $f_n\rightarrow f$ pointwise. Consider the following counterexample:

Let’s consider the series of functions the describe “moving triangles”. Let $f_1$ describe the triangle with base from $[0,1]$ and height 2 centered at $\frac{1}{2}$. Then the area of this triangle, or it’s integral, is 1. Let $f_2$ describe a triangle with base from $[0,1/2]$ and height 4 centered at $\frac{1}{4}$. Let $f_2$ be defined to be zero on $[1/2,1]$. Then the area of this triangle, and thus it’s integral, is also 1. We may continue in such a way, constructing triangles each with area 1 but with smaller and smaller bases on the x-axis. So, $f_n\rightarrow 0$ pointwise but $\int\limits_{0}^{1}f_n(x) dx=1$ for every $n\in\mathbb{N}$. Thus, $\lim\limits_{n\rightarrow\infty}\int\limits_{0}^{1}f_n(x) dx=1\not=0\int\limits_{0}^{1}0 dx$

Reflection: The proof part of this was not the hardest part for me. I hope that’s a good sign 🙂 The tricky part was coming up with the counter example. Thank you to Richard for sharing this one with me. It’s one of those nifty things to have up your sleeve.

Some other notes/thoughts about the exam. I am so ready to take this thing! I got the stomach flu this past Monday (5 days before the exam…ugh) and I was so upset, but today is my first day back to fully functioning life and I am happy to say that my analysis knowledge survived the flu! Now I am just ready to get this thing over! If anyone reading this has any last minute comments or analysis ideas they think I need to know, please feel free to share them in a comment on this post! I would be much appreciated 🙂

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One Response to Spring 2008 #5

1. Richard says:

The moving triangle, classic. You got this!